Eigenvalues of a Linear Transformation

We adopt the following convention: i, j, m, n denote natural numbers, K denotes a field, and a denotes an element of K. Next we state several propositions: (1) Let A, B be matrices over K, n1 be an element of Nn, and m1 be an element of Nm. If rng n1 × rngm1 ⊆ the indices of A, then Segm(A + B,n1,m1) = Segm(A,n1,m1) + Segm(B,n1,m1). (2) For every without zero finite subset P of N such that P ⊆ Seg n holds Segm(In×n K , P, P ) = I cardP×cardP K . (3) Let A, B be matrices over K and P , Q be without zero finite subsets of N. If P ×Q ⊆ the indices of A, then Segm(A+B,P,Q) = Segm(A,P,Q)+ Segm(B,P,Q).